When to use logistic regression and exact logistic regression

1. The general logistic regression process does not work very well for small sample set. The general logistic regression process is described here : http://www.ats.ucla.edu/stat/r/dae/logit.htm
2. For small sample sets, use exact logistic regression : http://www.ats.ucla.edu/stat/r/dae/exlogit.htm
3. To understand how the maximum likelihood estimation for logistic regression is biased for rare events, read : http://www.statisticalhorizons.com/logistic-regression-for-rare-events and http://www.cscu.cornell.edu/news/statnews/stnews82.pdf
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When to use exact logistic regression instead of regular logistic regression?

It is used when the sample size is too small for a regular logistic regression (which uses the standard maximum-likelihood-based estimator) and/or when some of the cells formed by the outcome and categorical predictor variable have no observations. The estimates given by exact logistic regression do not depend on asymptotic results.

What is separation in the data ? 

http://en.wikipedia.org/wiki/Separation_(statistics)

When there is separation in the data we use exact logistic regression or firths logistic regression ?

When the data is small use exact logistic regression. When you have a lot of non events then use firths logistic regression as suggested here http://sas-and-r.blogspot.com/2010/11/example-815-firth-logistic-regression.html

When do you use mixed effects logistic regression model ?
When there are fixed and random effects on the data. When the data has rank bias or kadu quality score bias, then you use mixed effects logistic regression. Some of the other biases might be variable and random. 
http://www2.hawaii.edu/~kdrager/MixedEffectsModels.pdf